VENN DIAGRAMS

In the last section we represented sets using set notation, listing the elements in brackets.With Venn Diagrams, we define everything exactly the same way, but the definitions are inthe form of pictures.

For instance, the set union of A and B, written A∪ B was defined as:

A∪ B = x / x ∈A or x ∈B{ }

With Venn Diagrams, the definition again is all the members that belong to A or B, but weshow that by shading in the circles A and B.

A U B

U

A B

That’s important, the set union is defined by shading in both circles of the Venn Diagram, allthe members belong.

Let’s look at the set intersection of A and B, that was defined as the members that belongedto both sets.

A∩ B = x / x ∈A and x ∈B{ }

That is illustrated by shading only the portion of the circles that overlap. Those elementsbelong to A and also to B.

A B

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Notice, there were rectangles around those examples. Those rectangles represented theuniversal set. All the elements under discussion.

To represent a single set, such as A. I would draw one circle and shade it in.

A

U

A

How would the complement of A, -A be illustrated? I’m glad you asked. Just like before, thecomplement of A are all the members of the universal set not in A.

A = x / x ∈ and x ∈ A{ }

That’s illustrated by shading in the rectangle, but not any part of the circle.

A

U

A

I know, you love coloring, you want to do more. Let’s look at the set difference. RememberA – B was defined as all the elements in A, but couldn’t belong to B.

A− B= x/ ∈A and x ∉B{ }

A – B

U

A B

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Having those operations defined through Venn Diagrams, we can now play with more sets.Again, the illustrations are important, introducing a third set using a circle does not in anyway change what we have already defined. In fact, if there is a third set, we work with justtwo at a time, just like we did with sets.

Example Shade A∩B∩C

First we’ll find the intersection of A and B, completely ignoring C. After that, we take thatresult and intersect it with C. We already know be definition, that A∩B looks like this

Now, we’ll take that shading and intersect that with circle C.

I shaded A∩B with horizontal lines and C with vertical so you can tell the difference.Where the shading overlaps is what these sets have in common. That almost makes sense,for an element to belong to A, B and C, there is only one region within the three circles thatsatisfies that.

A∩B∩C

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Let’s look at a Venn Diagram made up of three sets in which the regions are labeled. Now,we’ll describe each region.

U

A B

C

4 7

5

2

6

3 1

8

Region 5 is in all three circles. So any elements in region 5 would belong to all three sets.In other words; A∩B∩C .

What about Region 2? Those are the elements in A and B, but not C. How might youdescribe Region 6? Those are elements in B and C, but not in A. Try Region 4. Theelements in A and C, but not B.

This is fun, let’s look at some more regions. Region 1 describes the elements in A only.What about region 3? Those elements are only in B. Region 7 then would be the elementsin C only. Region 8 would describe elements that are not members of any of the sets, butbelong to the universal set.

It’s important that you become familiar with how each of those regions might be described.Being able to describe those regions would allow you to solve some problems.

Example

A survey was taken of 650 university students. It was reported that 240 were taking math,290 were taking biology, and 270 were enrolled in chemistry. Of those students, 80 weretaking biology and math, 70 were taking math and chemistry, 60 were taking biology andchemistry, and 50 were taking all three classes. How many students took math only?

At first glance, you might not think this is possible because the numbers add up to morethan 650. But if you are familiar with how the regions are described, we can determine howmany were in each region.

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In going about this problem, I would tell you to draw a Venn Diagram and begin by filling inRegion 5, the students that took all three courses.

U

M B

C

50

After doing that, we’ll place the number of students taking each course on the circle becausewe don’t know where those students should be located within the circles.

U

M B

C

50

290 240

270

Okay, now we can have some fun by determining what regions the students should belocated.

For instance, it says that 80 students are taking math and biology. We have 50 of thoseaccounted for in Region 5, how many does that leave to be in Region 2? 80 – 50 = 30That’s easy enough. Using that same reasoning, 70 students are taking math andchemistry, how many students would then be in Region 4? Well, 70 − 50 = 20 . That’spretty easy, don’t you think? Ok, how many students should be in Region 6? Since thereare 60 students enrolled in biology and chemistry, and 50 of them are accounted for inRegion 5, that leaves 10 students for Region 6.

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Let’s fill in those numbers:

U

M B

C

50

290 240

270

10 20

30

Now how many students would be in Region 1? Now remember, there are supposed to be240 students taking math, we have 100 accounted for in Regions 2, 4, and 5. That leaves140 students in region 1, taking math only. How many students are taking biology only?Well, we were told that 290 students were taking biology, we have 90 of them accounted forin Regions 2, 5, and 6, that leaves 200 students in Region 3. How many are taking onlychemistry? We know there are 270 students taking chemistry, we have 90 accounted for inRegions 4, 5, and 6, that leaves 190 in Region 7.

Filling in those numbers and taking the numbers off the circle, we have the followinginformation.

U

M B

C

50

290 240

270

10 20

30 200 140

190 10

We have one slight problem, if we add those regions within the circles, the total is 640students. The problem stated 650 were surveyed, we’re missing ten students. Where arethey? That’s right, they would be in Region 8, not taking any of those courses.

Now tell me, was that fun?

How many students took math and biology, but not chemistry?

How many students took math and chemistry, but not biology?

How many students took biology and chemistry, but not math?

How many students took only math?

How many students took exactly two of the courses?

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VENN DIAGRAMS

1a. A survey of 70 high school students revealed that 35 like folk music, 15 like classicalmusic, and 5 like both. How many of the students surveyed do not like either folk orclassical music?

2a. Out of 35 students in a finite math class, 22 are male, 19 are business majors, 27 arefreshmen, 14 are male business students, 17 are male freshmen, 15 are freshmenbusiness majors, and 11 are male freshmen business majors. How many upperclasswomen non-business majors are in the class? How many women business majorsare in the class?

3a. A survey of 100 college faculty who exercise regularly found that 45 jog, 30 swim, 20cycle, 6 jog and swim. 1 jogs and cycles, 5 swim and cycle, and 1 does all three.How many of the faculty members do not do any of these three activities? Howmany just jog?

4a. After a genetics experiment, the number of pea plants having certain characteristicswere tallied, with the results as follows.

22 were tall25 produce green peas39 produce smooth peas 9 are tall and produce green peas17 are tall and produce smooth peas.20 produce green peas and smooth peas 6 have all three characteristics 4 have none of the characteristics.

(a) Find the total number of plants counted.(b) How many plants are tall, but produce peas which are neither smooth nor green?(c) How many plants are not tall, but produce peas which are smooth and green?

5a. A survey of 80 business executives found the following recommendations on collegemajors for business students.

36 recommended liberal arts courses32 recommended business courses.32 recommended technical courses.16 recommended technical and business courses.16 recommended business and liberal arts courses.14 recommended liberal arts and technical courses 6 recommended all three.

(a) How many executives recommend liberal arts, but neither of the other two?(b) How many recommend none of these three types of courses?

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VENN DIAGRAMS

6a. The musical tastes of a number of college students were surveyed, and it was foundthat:

22 like Johnny Cash25 like Elvis Presley39 like The Carpenters 9 like Johnny Cash and Elvis Presley17 like Johnny Cash and The Carpenters20 like Elvis Presley and The Carpenters 6 like all three 4 like none of these performers

(a) How many students were surveyed?(b) How many like exactly two of these three performers?(c) How many like Johnny Cash only?(d) How many do not like Johnny Cash?

7a. The following data shows the preferences of 110 people at a wine-tasting party.

99 like Spanada96 like Ripple99 like Boone’s Fair Apple Wine95 like Spanada and Ripple94 like Ripple and Boone’s96 like Spanada and Boone’s93 like all three.

How many of the people:

(a) like none of the three beverages?(b) like Spanada, but not Ripple?(c) do not like Boone’s Farm Apple Wine?(d) like only Ripple?(e) like exactly two wines?

8a. Routine physical examinations of 500 pre-school children revealed that 40 haddental problems, 45 had vision problems, 55 had hearing problems, 15 had dentaland vision problems, 15 had dental and hearing problems, 20 had vision and hearingproblems, and 10 had dental, vision, and hearing problems. How many of thechildren had none of the three kinds of problems?

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VENN DIAGRAMS

9a. Human blood can contain either no antigens, the A antigen, the B antigen, or boththe A and B antigens. A third antigen, called the Rh antigen, is significant in humanreproduction, and again may or may not be present in an individual. Blood is calledtype A-positive if the subject has the A and Rh, but not the B antigen. Subjectshaving only the A and B antigens are said to have type AB-negative blood. Subjectshaving only the Rh antigen have type O-positive blood, etc. In a certain hospital thefollowing data on patients were recorded:

25 patients had the A antigen,17 had the A and B antigens,27 had the B antigen,22 had the B and Rh antigens,30 had the Rh antigen,12 had none of the antigens,16 had the A and Rh antigens,15 had all three antigens.

(a) How many patients are represented here?(b) How many patients have exactly one antigen?(c) How many patients have exactly two antigens?

10a. A local merchant uses television, radio, and newspaper advertising. To determinethe effectiveness of advertising, he questions 200 customers during a special after-hours sale to see how they knew about the sale. He found that 115 had seentelevision ads, 75 had heard radio ads, and 125 had read newspaper ads. He alsofound that 30 received information from television and radio, 70 from television andnewspapers, 25 from radio and newspapers, and 10 from all three. If everyone elsesaid they heard it from a friend, how many heard it from a friend?

11a. In order to prepare a report on agricultural prospects for his county, the county farmadvisor questions 100 farmers about their crop plans for the following year. He findsthat 75 intend to plant corn, 55 will plant soybeans, 35 will plant wheat, 35 will plantcorn and soybeans, 25 will plant corn and wheat, 15 will plant soybeans and wheat,and 10 will plant all three. How many of the farmers will plant only one crop? Howmany will plant at least two crops?

12a. In a group of 150 primary students, 100 watch “Sesame Street,” 55 watch “ElectricCompany,” and 65 watch “Mr. Rogers’ Neighborhood.” If, in addition, 35 watch“Sesame Street” and “Electric Company,” 45 watch “Sesame Street” and “Mr.Rogers,” 30 watch “Electric company” and “Mr. Rogers,” and 20 watch all three, howmany watch none of the three? How many watch only “Sesame Street”?

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VENN DIAGRAMS

13a. At a pow-wow in Arizona, Native Americans from all over the Southwest came toparticipate in the ceremonies. A coordinator of the pow-wow took a survey andfound that:

15 families brought food, costumes, and crafts:25 families brought food and crafts:42 families brought food:20 families brought costumes and food: 6 families brought costumes and crafts, but not food: 4 families brought crafts, but neither food nor costumes:10 families brought none of the three items:18 families brought costumes, but not crafts:

(a) How many families were surveyed?

(b) How many families brought costumes?

(c) How many families brought crafts, but not costumes?

(d) How many families did not bring crafts?

(e) How many families brought food or costumes?

14a. A survey of people attending a Lunar New Year celebration in Chinatown yielded thefollowing results:

120 were women:150 spoke Cantonese:170 lit firecrackers:108 of the men spoke Cantonese:100 of the men did not light firecrackers: 18 of the non-Cantonese-speaking women lit firecrackers: 78 non-Cantonese-speaking men did not light firecrackers: 30 of the women who spoke Cantonese lit firecrackers.

(a) How many attended?

(b) How many of those who attended did not speak Cantonese?

(c) How many women did not light firecrackers?

(d) How many of those who lit firecrackers were Cantonese-speaking men?

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VENN DIAGRAMS

15a. A chicken farmer surveyed his flock with the following results. The farmer had:

9 fat red roosters: 2 fat red hens:37 fat chickens:26 fat roosters 7 thin brown hens:18 thin brown roosters: 6 thin red roosters: 6 thin red hens:

Answer the following questions about the flock. [Hint: You need a Venn Diagram withregions for fat, for male (a rooster is a male, a hen is a female), and for red (assume thatbrown and red are opposites in the chicken world).] How many chickens were:

(a) fat? (b) red?

(c) male? (d) fat, but not male?

(e) brown, but not fat? (f) red and fat?

16. Country-Western songs seem to emphasize three basic themes: love, prison, andtrucks. A survey of the local country-western radio station produced the followingdata:

12 songs were about a truck driver who was in love while in prison:13 were about a prisoner in love:28 were about a person in love:18 were about a truck driver in love. 3 were about a truck driver in prison who was not in love: 2 were about a prisoner who was not in love and did not drive a truck:8 were about a person who was not in prison, not in love, and did not

drive a truck:16 were about truck drivers who were not in prison.

(a) How many songs were surveyed?

Find the number of songs about:

(b) truck drivers: (c) prisoners:

(d) truck drivers in prison: (e) people not in prison:

(f) people not in love:

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VENN DIAGRAMS

17A. A survey of 80 sophomores at a western college showed that:

36 took English:32 took history:32 took political science:16 took political science and history:16 took history and English:14 took political science and English: 6 took all three

How many students:

(a) took English and neither of the other two?

(b) took none of the three courses?

(c) took history, but neither of the other two?

(d) took political science and history, but not English?

(e) did not take political science?

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